Theory of electron capture from a hydrogenlike ion by a bare ion

Abstract

We have systematically developed a new approximation for the calculation of the cross section for electron capture from a hydrogenlike ion of large nuclear charge $Z_{T}e$ by a bare ion of charge $Z_{P}e$ moving with speed $v$. The amplitude in the wave treatment is obobtained through consistent expansion in the small parameters $\frac{Z_P}{Z_T}$ and $\frac{Z_P{e}^2}{\hslash v}$; however, the ratio $\frac{Z_T{e}^2}{\hslash v}$ is not assumed small. Following a careful analysis of the approach to the energy shell, an off-shell factor is seen to arise which does not appear in the impulse approximation. The effects of this factor on the capture amplitude are explored. Using a peaking approximation, we derive a closed-form expression for the $1s\to \mathrm{ns}$ capture amplitude which includes the effects of the off-shell factor and is accurate to order ${\left(\frac{Z_P{e}^2}{\hslash v}\right)}^2$. We tabulate the peaking-approximation cross section. The derivation of the asymptotic form of the cross section when $\frac{Z_P}{Z_T}<<1\mathrm{}$ is justified. Using a realistic one-electron model, $K$-shell capture cross sections are calculated for protons on carbon, neon, and argon. These are seen to represent the data generally better than do the impulse approximation cross sections.